Nonlinear analysis with an arbitrary stimulus ensemble
Authors:
Jonathan D. Victor and Bruce W. Knight
Journal:
Quart. Appl. Math. 37 (1979), 113-136
MSC:
Primary 92A09
DOI:
https://doi.org/10.1090/qam/542986
MathSciNet review:
542986
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Abstract: A family of Wiener-type methods is discussed in a general context. These methods share the concept of expansion of an unknown transducer as an orthogonal series. The terms of the series are drawn from a hierarchy of subspaces of transducers that are orthogonal with respect to a particular stimulus ensemble. Choices of specific stochastic ensembles lead to previously described analytical methods, including the classical one of Wiener.
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S. E. Brodie, B. Knight, and F. Ratliff, The response of the Limulus retina to moving stimuli: a prediction by Fourier synthesis, J. Gen. Physiol. 72, 129–166 (1978)
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A. S. French and R. K. S. Wong, Nonlinear analysis of sensory transduction in an insect mechanoreceptor, Biol. Cybernetics 26, 231–240 (1977)
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V. Z. Marmarelis, A family of quasi-white random signals and its optimum use in biological system identification I. Theory, Biol. Cybernetics 27, 49–56 (1977)
G. D. McCann, Nonlinear identification theory models for successive stages of visual nervous systems of flies, J. Neurophysiol. 37, 869–895 (1974)
A. R. Møller, Statistical evaluation of the dynamic properties of cochlear nucleus units using stimuli modulated with pseudorandom noise, Brain Res. 57, 443–456 (1973)
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M. Abramowitz and I. Stegun, eds., Handbook of mathematical functions, Dover, N.Y., 1959
J. F. Barrett, The use of functionals in the analysis of nonlinear physical systems, J. Electronics and Control 15, 567–615 (1963)
E. Bedrosian and S. O. Rice, The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs, Proc. IEEE 59, 1688–1707 (1971)
A. S. Besicovitch, Almost periodic functions, Dover, N.Y., 1954
M. Biederman-Thorson and J. Thorson, Dynamics of excitation and inhibition in the light-adapted Limulus eye in situ, J. Gen. Physiol. 58, 1–19 (1971)
H. W. Bode, Network analysis and feedback amplifier design, D. van Nostrand Company, N.Y., 1945
A. Borsellino, R. E. Poppele, and C. A. Terzuolo, Transfer functions of the slowly adapting stretch receptor organ of Crustacea, Cold Spring Harbor Symp. Quant. Biol. 30, 581–586 (1965)
S. E. Brodie, B. Knight, and F. Ratliff, The response of the Limulus retina to moving stimuli: a prediction by Fourier synthesis, J. Gen. Physiol. 72, 129–166 (1978)
S. E. Brodie, B. Knight, and F. Ratliff, The spatiotemporal transfer function of the Limulus lateral eye, J. Gen. Physiol. 72, 167–202 (1978)
R H. Cameron and W. T. Martin, The orthogonal development of nonlinear functionals in a series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)
J. Cooley and J. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Computation 19, 297–301 (1965)
H. deLange, Attenuation characteristics and phase-shift characteristics of the human fovea-cortex systems in relation to flicker-fusion phenomena, Doctoral dissertation, Technische Hogeschool, Delft, 1957
R. DeVoe, Linear superposition of retinal action potentials to predict electrical flicker responses from the eye of the wolf spider, Lycosa baltimoriana (Keyserling), J. Gen. Physiol. 46, 75–96 (1962)
F. Dodge, R. Shapley, and B. Knight, Linear systems analysis of the Limulus retina, Behav. Sci. 15, 24–36 (1970)
H. Duifhuis, Cochlear nonlinearity and second filter: possible mechanism and implications, J. Acoust. Soc. Am. 59, 408–423 (1976)
J. F. Fohlmeister, R. E. Poppele, and R L. Purple, Repetitive firing: a quantitative study of feedback in model encoders. J. Gen. Physiol. 69, 815–848 (1977)
J. F. Fohlmeister, R. E. Poppele, and R. L. Purple, Repetitive firing: quantitative analysis of encoder behavior of slowly adapting stretch receptor of crayfish and eccentric cell of Limulus, J. Gen. Physiol. 69, 849–877 (1977)
A. S. French, Practical nonlinear system analysis by Wiener kernel estimation in the frequency domain, Biol. Cybernetics 24, 111–119 (1976)
A. S. French and R. K. S. Wong, Nonlinear analysis of sensory transduction in an insect mechanoreceptor, Biol. Cybernetics 26, 231–240 (1977)
K. Fukurotani, K.-I. Hara, and Y. Oomura, Dynamic characteristics of the receptive field of L-cells in the carp retina, Vision Res. 15, 1403–1405 (1975)
E. Hille and R. S. Phillips, Functional analysis and semigroups, American Mathematical Society, Providence, R.I., 1957
S. Hochstein and R. Shapley, Linear and nonlinear spatial submits in Y cat retinal ganglion cells, J. Physiol. 262, 265–284 (1976)
G. W. Hughes and L. M affei, Retinal ganglion cell response to sinusoidal light stimulation, J. Neurophysiol. 29, 333–352 (1966)
H. E. Ives, A theory of intermittent vision, J. Opt. Soc. Am. 6, 343–361 (1922)
S. Klein and S. Y asui, Nonlinear systems analysis with non-Gaussian white stimuli: general basis functionals and kernels, IEEE Trans. Inf. Theory, in press (1979)
B. W. Knight, J.-I. Toyoda, and F. A. Dodge, A quantitative description of the dynamics of excitation and inhibition in the eye of Limulus, J. Gen. Physiol. 36, 421–437 (1970)
H. I. Krausz, Identification of nonlinear systems using random impulse trains, Biol. Cybernetics 19, 217–230 (1975)
H. I. Krausz and W. O. Friesen, The analysis of nonlinear synaptic transmission, J. Gen. Physiol. 70, 243–265 (1977)
Y. N. Lee and M. Schetzen, Measurement of the kernels of a nonlinear system by cross-correlation, Int. J. Control 2, 237–254 (1965)
E. D. Lipson, White noise analysis of Phycomyces light growth response system I., II., III, Biophys. J. 15, 989–1045 (1975)
G. Marchesini and G. Picci, Sull’ identificazione funzionale di sistemi nonlinearri in regime periodico, Rendicotti dell’ A.E.I. (1969)
P. Z. Marmarelis and K.-I. Naka, White noise analysis of a neuron chain: An application of the Wiener theory. Science 175, 1276–1278 (1972)
P. Z. Marmarelis and K.-I. Naka, Nonlinear analysis of receptive field responses in the catfish retina, II. One-input white noise analysis, J. Neurophysiol. 36, 619–633 (1973)
V. Z. Marmarelis, A family of quasi-white random signals and its optimum use in biological system identification I. Theory, Biol. Cybernetics 27, 49–56 (1977)
G. D. McCann, Nonlinear identification theory models for successive stages of visual nervous systems of flies, J. Neurophysiol. 37, 869–895 (1974)
A. R. Møller, Statistical evaluation of the dynamic properties of cochlear nucleus units using stimuli modulated with pseudorandom noise, Brain Res. 57, 443–456 (1973)
K.-I. Naka, P. Z. Marmarelis, and R. Y. Chan, Morphological and functional identification of catfish retinal neurons, III. Functional identification, J. Neurophysiol. 38, 92–131 (1975)
G. Palm and T. Poggio, The Volterra representation and the Wiener expansion: validity and pitfalls, SIAM J. Appl. Math., 33, 195–216 (1977)
G. Palm and T. Poggio, Stochastic identification methods for nonlinear systems: An extension of the Wiener theory, personal communication (1977)
R. B. Pinter, Sinusoidal and delta function responses of visual cells of the Limulus eye, J. Gen. Physiol. 49, 565–593 (1966)
R. A. Price, A useful theorem for nonlinear devices having Gaussian inputs, IRE Trans. Information Theory IT-4, 69–72 (1958)
J. W. S. Pringle and V. J. Wilson, The response of a sense organ to a harmonic stimulus, J. Exp. Biol. 29, 220–234 (1952)
R. L. Purple and F. A. Dodge, Interaction of excitation and inhibition in the eccentric cell in the eye of Limulus, Cold Spring Harbor Symp. Quant. Biol. 30, 529–537 (1965)
F. Ratliff, B. Knight, F. Dodge, and H. K. Hartline, Fourier analysis of dynamics of excitation and inhibition in the eye of Limulus: amplitude, phase, and distance, Vision Res. 14, 1155–1168 (1974)
A. Sandberg and L. Stark, Wiener G-functional analysis as an approach to nonlinear characteristics of human pupil light reflex, Brain Res. 11, 194–211 (1968)
R. Shapley and J. Victor, The effect of contrast on the transfer properties of cat retinal ganglion cells, J. Physiol. 285, 275–298 (1978)
L. Sirovich and I. Abramov, Photopigments and pseudopigments. Vision Res. 17, 5–16 (1977)
H. Spekreijse, Rectification in the goldfish retina: analysis by sinusoidal and auxiliary stimulation, Vision Res. 9, 1461–1472 (1969)
H. Spekreijse, O. Estevez, and D. Reits, Visual evoked potentials and the physiological analysis of visual processes in man, in J. E. Desmedt (ed.), Visual evoked potentials in man. Clarendon Press, Oxford, 1977
G. J. St.-Cyr and D. H. Fender, Nonlinearities of the human oculomotor system: gain. Vision Res. 9, 1235–1246 (1969)
G. Szego, Orthogonal polynomials, American Mathematical Society, Providence, R. I., 1939
J. Thorson and M. Biederman-Thorson, Distributed relaxation processes in sensory adaptation, Science 183, 161–172 (1974)
J. Victor, Nonlinear systems analysis: comparison of white noise and sum of sinusoids in a biological system, Proc. Nat. Acad. Sci. USA 76, 996–998 (1979)
J. Victor and R. Shapley, A method of nonlinear analysis in the frequency domain, Biophys. J., submitted (1979)
J. Victor, R. Shapley, and B. Knight, Nonlinear analysis of cat retinal ganglion cells in the frequency domain, Proc. Nat. Acad. Sci. USA 74, 3068–3072 (1977)
V. Volterra, Theory of functionals and of integral and integro-differential equations, London and Glasgow: Blackie and Sons, N.Y.: Dover, N.Y., 1959
W. von Seelen and K. P. Hoffman, Analysis of neuronal networks in the visual system of the cat using statistical signals, Biol. Cybernetics 22, 7–20 (1976)
N. Wiener, Nonlinear problems in random theory, M.I.T. Press and Wiley, N.Y., 1965
V. J. Wilson, B. Peterson, K. Fukushima, N. Hirai, and Y. Uchino, Analysis of vestibulocollic reflexes by sinusoidal polarization of vestibular afferent fibers, J. Neurophysiol. 42, 331–346 (1979)
S. Yasui, Stochastic functional Fourier series, Volterra series, and nonlinear systems analysis, IEEE Trans. Auto. Control, in press (1979)
L. R. Young and L. Stark, Variable feedback experiments testing a sampled data model for eye tracking movements, IEEE Trans. HFE-4, 38–51 (1963)
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© Copyright 1979
American Mathematical Society
